TRIQ: a new method to evaluate triclusters

Abstract

Background

Triclustering has shown to be a valuable tool for the analysis of microarray data since its appearance as an improvement of classical clustering and biclustering techniques. The standard for validation of triclustering is based on three different measures: correlation, graphic similarity of the patterns and functional annotations for the genes extracted from the Gene Ontology project (GO).

Results

We propose TRIQ, a single evaluation measure that combines the three measures previously described: correlation, graphic validation and functional annotation, providing a single value as result of the validation of a tricluster solution and therefore simplifying the steps inherent to research of comparison and selection of solutions. TRIQ has been applied to three datasets already studied and evaluated with single measures based on correlation, graphic similarity and GO terms. Triclusters have been extracted from this three datasets using two different algorithms: TriGen and OPTricluster.

Conclusions

TRIQ has successfully provided the same results as a the three single evaluation measures. Furthermore, we have applied TRIQ to results from another algorithm, OPTRicluster, and we have shown how TRIQ has been a valid tool to compare results from different algorithms in a quantitative straightforward manner. Therefore, it appears as a valid measure to represent and summarize the quality of tricluster solutions. It is also feasible for evaluation of non biological triclusters, due to the parametrization of each component of TRIQ.

Background

Analysis of data structured in 3D manner is becoming an essential task in fields such as biomedical research, for instance in experiments studying gene expression data taking time into account. There is a lot of interest in this type of longitudinal experiments because they allow an in-depth analysis of molecular processes in which the time evolution is important, for example, cell cycles, development at the molecular level or evolution of diseases [1]. Therefore, the use of specific tools for data analysis in which genes are evaluated under certain conditions considering the time factor becomes necessary. In this sense, triclustering [2] appears as a valuable tool since it allows for the assessment of genes under a subset of the conditions of the experiment and under a subset of time points.

The evaluation of solutions obtained by triclustering algorithms is challenging by the fact that there is no ground truth to describe triclusters present in real 3D data. In literature, the standard measures to evaluate tricluster solutions are based on three areas as can be seen in the triclustering publications [3–7]. First, correlation measures such as Pearson [8] or Spearman [9]. Second, graphic validation of the patterns extracted based on the graphic representation, i.e., how similar the genes from a tricluster are based on the graphic representation of the genes across conditions and time points. Third, functional annotations extracted from the Gene Ontology project (GO) [10] for the genes in the tricluster.

However, we consider that providing a single evaluation measure capable of combining the information from the three aforementioned sources of validation is a neccesary task. Therefore, in this work we propose TRIQ, a validation measure which combines the three previously proposed validation mechanisms (correlation, graphic validation and functional annotation of the genes).

The application of clustering and biclustering techniques to gene expression data has been broadly studied in the literature [11, 12]. Although triclustering is the result from the natural evolution of the clustering and biclustering techniques, is still a very recent concept. However, nowadays, these techniques are arousing a great interest from the scientific community, which has caused a notable increase of the number of researches focused on finding new triclustering approaches. This section is to provide a general overview of triclustering published in literature. We particularly focus on the validation methods applied to assess the quality of the triclusters obtained.

In 2005, Zhao and Zaki [3] introduced the triCluster algorithm to extract patterns in 3D gene expression data. They presented a measure to assess triclusters’s quality based on the symmetry property. They validated their triclusters based on their graphical representation and Gene Ontology (GO) results. g-triCluster, an extended and generalized version of Zhao and Zaki’s proposal, was published one year later [4]. The authors claimed that the symmetry property is not suitable for all patterns present in biological data and proposed the Spearman rank correlation [9] as a more appropriate tricluster evaluation measure. They also showed validation results based on GO.

An evolutionary computation proposal was made in [13]. The fitness function defined is a multi-objective measure which tries to optimize three conflicting objectives: clusters size, homogeneity and gene-dimension variance of the 3D cluster. The tricluster quality validation was based on GO. LagMiner was introduced in [6] to find time-lagged 3D clusters, what allows to find regulatory relationships among genes. It is based on a novel 3D cluster model called S₂D₃ Cluster. They evaluated their triclusters on homogeneity, regulation, minimum gene number, sample subspace size and time periods length. Their validation was based on graphical representation and GO results. Hu et al. presented an approach focusing on the concept of Low-Variance 3-Cluster [5], which obeys the constraint of a low-variance distribution of cell values. This proposal uses a different functional enrichment tool called CLEAN [14], which uses GO as one of their components. The work in [7] was focused on finding Temporal Dependency Association Rules, which relate patterns of behavior among genes. The rules obtained are used to represent regulated relations among genes. They also validated their triclusters based on their graphical representation and GO results.

Tchagang et al. [15] proposed OPTricluster, a triclustering algorithm which obtains 3D short time series gene expression datasets by applying a statistical methodology. In this case, the authors carried out an in-depth biological validation based on GO, but they tested the robustness of OPTricluster to noise using the Adjusted Rand Index (ARI) [16], which also was used by aforementioned g-tricluster.

In 2013, two new and very interesting approaches were proposed. On the one hand, the δ−TRIMAX algorithm [17], which applies a variant of the MSR adapted to 3D datasets and yields triclusters that have a MSR score below a threshold δ. This algorithm has a version based on evolutionary multi-objective optimization, named EMOA−δ−TRIMAX [18], which aims at optimizing the use of δ−TRIMAX by adding the capabilities of evolutionary algorithms to retrieve overlapping triclusters. On the other hand, OAC-Triclustering was also proposed by Gnatyshak et al. in [19]. In the following years, the authors developed improvements and extensions of this algorithm [20–22].

More recent works have extended the capabilities of the tricluster algorithms by combination of several approaches. Thereby, Liu et al. [23] mixed fuzzy clustering and fuzzy biclustering algorithms in order to expands them to support 3D data and they used the F-Measure and Entropy as criteria to evaluate the performance. Also, Kakati et al. [24] combined parallel biclustering and distributed triclustering approaches to obtain improvements on the computational cost. In this work, the authors use a quality measure based on shifting and scaling patterns [25] to optimize the triclusters obtained.

Most of the methods studied base the quality of the triclusters on the graphic representation or on metrics aimed at measuring diverse characteristics of such representation. From a biological point of view, the standard for validation of triclusters quality is based on GO functional annotations.

Methods

This section presents the TRIQ (TRIcluster Quality) validation measure [26], a novel method to evaluate the quality of triclusters extracted from gene expression datasets.

From an overall perspective, TRIQ takes into account the three principal components of a tricluster, i.e. the genes, experimental conditions and time points, in order to measure its quality from three approaches: the level of biological notoriety of the cluster (biological quality), the graphic quality of the patterns of the genes in the tricluster (graphic quality), and the level of correlation of the genes in the tricluster by means of the Pearson [8] and the Spearman [9] indexes. Therefore, TRIQ is composed by a combination of four indexes: BIOQ (BIOlogical Quality), GRQ (GRaphic Quality), PEQ (PEarson Quality) and SPQ (SPearman Quality).

In Eq. 1 we define TRIQ as the weighted sum of each of the four aforementioned terms. Therefore, four associated weights must be defined: the weight for BIOQ, denoted as W_bio; the weight for GRQ, denoted as W_gr; the weight for PEQ, denoted as W_pe; and the weight for SPQ, denoted as W_sp.

$$ \begin{aligned} TRIQ(TRI) &= \frac{1}{W_{bio}+W_{gr}+W_{pe}+W_{sp}} * \left[ W_{bio}*BIOQ(TRI) \right.\\ &\left.+ W_{gr}*GRQ(TRI) + W_{pe}*PEQ(TRI) + W_{sp}*SPQ(TRI)\right] \\ \end{aligned} $$

(1)

This is a general definition of TRIQ. In order to obtain a TRIQ index as balanced as possible among the four quality indexes BIOQ, GRQ, PEQ, and SPQ we performed an exhaustive testing procedure with well known datasets. Several combinations of values of BIOQ, GRQ, PEQ, and SPQ were tested, and in Fig. 1 we show the results obtained.

Fig. 1

Representation of BIOQ, GRQ, PEQ, and SPQ influence on TRIQ

We see that that the value of TRIQ is slightly directly dependent on the weights related to correlation, PEQ, and SPQ. This is due to the fact that these values rank in the [0-1] interval, being usually high, from 0.7 to 1. The value of TRIQ has a higher level of dependence to the graphical quality, GRQ, and reverse strong dependence to the biological quality, BIOQ, due to the fact that BIOQ ranks in low values, usually around 10⁻³ to 10⁻⁵. Based on this experiments, we have configured the TRIQ measure with the weights showed in Eq. 2 in order to obtain a balanced value of TRIQ.

$$ W_{bio} = 0.5, W_{gr} = 0.4, W_{pe} = 0.05, W_{sp} = 0.05 $$

(2)

Next, we describe in depth each of the terms involved in the TRIQ measure.

Correlation measures: PEQ and SPQ

The correlation measures involved in TRIQ are Pearson’s PEQ [8] and Spearman’s SPQ [9] correlations. They have been chosen since they are the standard correlation measures and they are widely used in literature [4]. The correlation provides a numerical estimation of the dependence among the genes, conditions and times in the tricluster solutions.

Given a tricluster TRI, we compute PEQ and SPQ by the following mechanism. Given the subset of genes (see Eq. 3a), conditions (see Eq. 3b) and time stamps (see Eq. 3c), we obtain a value of expression for each combination gene, condition and time. For instance, for a tricluster consisting of four genes, two conditions and three time points, we have twenty four expression values. We then compute the Pearson correlation for each pair of values, and compute PEQ as the average of the absolute values to avoid negative and positive correlations canceling each other (see Eq. 4). Furthermore, for this measure we do not care if the correlation is positive or negative between values, we only want to know the level of correlation. The SPQ value is the equivalent using the Spearman correlation (see Eq. 5).

$$\begin{array}{@{}rcl@{}} TRI_{G} &=& <g_{0}, g_{1}, \ldots, g_{G}>\end{array} $$

(3a)

$$\begin{array}{@{}rcl@{}} TRI_{C} &=& <c_{0}, c_{1}, \ldots, c_{C}>\end{array} $$

(3b)

$$\begin{array}{@{}rcl@{}} TRI_{T} &=& <t_{0}, t_{1}, \ldots, t_{T}>\end{array} $$

(3c)

$$ PEQ(TRI) = \frac{\sum_{i=0,j=0}^{\#exp} \left|Pearson_{i \neq j}\left(exp_{i}, exp_{j}\right)\right|}{\#pairs\;of\;exp} $$

(4)

$$ SPQ(TRI) = \frac{\sum_{i=0,j=0}^{\#exp} \left|Spearman_{i \neq j}\left(exp_{i}, exp_{j}\right)\right|}{\#pairs\;of\;exp} $$

(5)

with exp representing the expressions in each tricluster TRI.

Graphical validation: GRQ

The GRQ member of Eq. 1 measures the graphical quality of the tricluster. This graphical quality of a tricluster is a quantitative representation of a qualitative measure: how homogeneous the members of the tricluster are. This method is widely used in literature for visual validation of the results by means of graphically representing the triclusters on their three components: genes, conditions and time points [3, 6, 7].

The GRQ index is described in Eq. 6. This measure is defined based on the normalization of the angle value given by MSL. The Multi SLope (MSL) evaluation function was defined in [27] and, given a tricluster TRI, provides a numerical value of the similarity among the angles of the slopes formed by each profile shaped by the genes, conditions, and times of the tricluster.

$$ GRQ(TRI) = 1 - \frac{MSL(TRI)}{2\pi} $$

(6)

The MSL measure considers the three graphical views of a tricluster, also defined in [27]: TRI_gct, TRI_gtc, and TRI_tgc. These three terms are generally defined as TRI_xop, with the expression levels of the tricluster represented in the Y axis, x represents the tricluster component in the X axis (genes or time points), o represents the lines plotted in the graph (genes, conditions or time lines) and p the type of facets or panels represented (time points or conditions). We can observe an example of the TRI_tgc view of a tricluster with the genes g₁, g₄, g₇ and g₁₀, the experimental conditions c₂, c₅ and c₈ and the time points t₀, t₂, t₁₁ in Fig. 2 and see how each line or gene forms a set of angles (two for this particular example) defined by each time point in the X axis for every panel or experimental condition. Thus, MSL measures the differences among the angles formed by every series traced on each of the three graphic representations taking into account TRI_gct, TRI_gtc, and TRI_tgc. A near to zero value of MSL implies a better graphical quality of a tricluster therefore, according to GRQ formulation in Eq. 6, a tricluster is graphically better the smaller the value of MSL.

Fig. 2

Representation of how the MSL measure is calculated. This figure shows three graphics containing four lines each one with a representation of their slopes

Biological validation: BIOQ

The BIOQ member of Eq. 1 measures the biological quality of the tricluster. Specifically, BIOQ uses the genes (TRI_G) of the input tricluster TRI to compute this index. As you can see in Eq. 7, the biological quality of a tricluster TRI is defined as the biological significance, SIG_bio, of the set of genes TRI_G divided by the S_max value.

$$ BIOQ(TRI) = \frac{SIG_{bio}\left(TRI_{G}\right)}{S_{max}} $$

(7)

The SIG_bio and S_max elements of the BIOQ index have been designed in order to represent, by means of a quantitative score, the value of the Gene Ontology analysis of the genes that compose the measured tricluster.

The Gene Ontology Project (GO) [10] is a major bioinformatics initiative with the aim of standardizing the representation of gene and gene product attributes across species and databases, besides identifying the annotated terms, performs the statistical analysis for the over-representation of those terms, also providing a statistical significance p-value. However, it is also important to take into account how deep in the ontology the terms are annotated, with the deeper terms being more specific than the superficial ones [28]. The SIG_bio and S_max elements are calculated based on the GO analysis that identifies, for a set of genes in a tricluster, the terms listed in each of the three available ontologies: biological processes, cellular components, and molecular functions. This GO analysis is performed with the software Ontologizer [29].

The computation of SIG_bio consists on counting how many terms of the annotated genes of the tricluster in the GO analysis are in a particular intervals of p-value. Table 1 represents the ah-hoc designed system of intervals of p-value and scoring system. The intervals and the scoring system are defined in Eq. 8 where for a given level, Inter_l is defined by a weight value w_l for the level, and by the lower and upper bounds (inf_l and sup_l, respectively), being an open-closed p-values interval (Eq. 8a). The set of existing LV consists of all levels with Inf_l smaller or equal to a minimum p-value, th. For each interval of each level Inter_l, the weight value w_l is defined in Eq. 8c; Inf_l is defined in Eq. 8d, and sup_l is defined in Eq. 8e.

Table 1 Biological significance intervals

This definition is made in order to establish a general interval system dependent on the parameters described above. For our purpose, we have settled these parameters as shown in Eq. 9; this configuration produces the intervals of Table 1, furthermore, it describes all the biological significance intervals for the configuration detailed in Eq. 9. For each row, weight (wl) and range (inter_l) for each level (L) sorted in ascending order are shown. Each interval provides a set of p-values where their significance is directly related to the corresponding level, that is, a p-value is better the higher the level to which it belongs, and a p-value is better the closer to zero it is.

$$\begin{array}{@{}rcl@{}} th &=& 1.0\times 10^{-40} \end{array} $$

(9a)

$$\begin{array}{@{}rcl@{}} d &=& 10.0 \end{array} $$

(9b)

$$\begin{array}{@{}rcl@{}} b &=& 10.0 \end{array} $$

(9c)

$$\begin{array}{@{}rcl@{}} s &=& 1.0 \end{array} $$

(9d)

$$\begin{array}{@{}rcl@{}} LV &=& \lbrace 1,\ldots,41 \rbrace \end{array} $$

(9e)

Taking into account each level l and each predefined interval inter_l, the biological significance for the genes of the measured tricluster is defined in Eq. 10a as the addition of all scores for each level l from the LV level set Eq. 9e. The score function S for a level l (Eq. 10b) is defined by the multiplication of the concentration of terms for this level C(l), defined in Eq. 10c as the number of terms of the level l divided by the total number of terms, by the weight of the level, and by the level plus a bonus function f_bonus, defined in Eq. 10d as the sum of the level plus a bonus value V_bonus if the current level is the maximum level of LV or zero in any another case.

$$\begin{array}{@{}rcl@{}} SIG_{bio} (TRI_{G}) &=& \sum_{l \in LV} S(l) \end{array} $$

(10a)

$$\begin{array}{@{}rcl@{}} S(l) &=& C(l)*w_{l}*l + f_{bonus}(l) \end{array} $$

(10b)

$$\begin{array}{@{}rcl@{}} C(l) &=& \frac{\#terms(l)}{\#total\;terms} \end{array} $$

(10c)

$$\begin{array}{@{}rcl@{}} f_{bonus}(l) &=& if\; \left(l \;equal\; to\; l_{max}\right)\; then\;\; l + V_{bonus}\;\; else 0 \end{array} $$

(10d)

Again, this definition is made in order to establish a general system of SIG_bio. For our purpose and as a result of an exhaustive testing, the V_bonus parameter has been settled to 0; this fact produces S_max as the maximum achievable score for the interval configuration as you can see in Eq. (11), that has been used to the SIG_bio normalization in Eq. 7.

$$ S_{max} = \left(C_{max} * w_{l_{41}} * l_{41}\right) + f_{bonus}(41) = (1 * 401 * 41) + (41 + 0) = 16482 $$

(11)

Results

In this section, we present how TRIQ works in an experimental environment. To reach this goal, we have used the TriGen algorithm [2] and the OPTricluster algorithm [15] in order to analyze the datasets, find triclusters and measure them with TRIQ.

TriGen is based on an heuristic, genetic algorithm, and its performance greatly depends on the fitness function used to find the triclusters. There are three fitness functions available in TriGen: Mean Squared Residue 3D (MSR_3D) [30], Least Squared Lines (LSL) [31] and Multi SLope Measure (MSL) [27]. OPTricluster identifies triclusters of genes with expression levels having the same direction across the time point experiments in subsets of samples taking into consideration the sequential nature of the time-series.

The three datasets analyzed that involve genes and experimental conditions examined under certain time points are:

• $D_{elu_{3D}}$: The yeast cell cycle (Saccharomyces Cerevisiae) [32], in particular, the elutriation experiment.

• $D_{GDS4510_{3D}}$: The GDS4510 dataset from an experiment with mice (Mus Musculus) [33].

• $D_{GSD4472_{3D}}$: The GDS4472 dataset from an experiments with humans (Homo Sapiens) [34].

The first dataset is available at the Stanford University website. The last two datasets have been retrieved from Gene Expression Omnibus [35], a repository of high throughput gene expression data.

For each dataset, we have performed four algorithm executions: TriGen with MSR_3D (hereon MSR_3D), TriGen with LSL (hereon LSL), TriGen with MSL and (hereon MSL) and OPTricluster (hereon OPT).

For each algorithm execution and dataset, we have yielded 10 triclusters and the TRIQ measure has been used to evaluate their quality. We have found 10 triclusters for each execution in order to have a high number of solutions where TRIQ can show its suitability. In the case of MSR_3D, LSL, and MSL executions the number of triclusters has been chosen as one of the TriGen algorithm parameters and for OPT executions, the tricluster have been randomly selected from the wide collection of triclusters yielded.

Summarizing, we present three experimental batches (Yeast Elutriation Dataset, Mouse GDS4510 Dataset and, Human GDS4472 Dataset) with four experiments each one: MSR_3D, LSL, MSL and OPT.

Yeast elutriation dataset

This batch corresponds to the yeast (Saccharomyces Cerevisiae) cell cycle problem [32]. The yeast cell cycle analysis project’s goal is to identify all genes whose mRNA levels are regulated by the cell cycle. The resources used are public and available in http://genome-www.stanford.edu/cellcycle/. There, we can find information relative to gene expression values obtained from different experiments using microarrays.

For our purpose, we have created a dataset $D_{elu_{3D}}$ from the elutriation experiment with 7744 genes, 13 experimental conditions, and 14 time points. Experimental conditions correspond to different statistical measures of the Cy3 and Cy5 channels while time points represent different moments of taking measures from 0 to 390 min.

$D_{elu_{3D}}$ has been used as the input of the TriGen and the OPTtricluster algorithm in four experiments: MSR_3D, LSL, MSL and, OPT.

Elutriation M S R _3D experiment

We can verify in Table 2 how TRI₉ has the best values of BIOQ, PEQ and SPQ whereas TRI₁₀ has the best value of GRQ. The GRQ, PEQ and SPQ values are stabilized from TRI₂ to TRI₈ until TRI₉−TRI₁₀ when these values reach the maximum. Regarding BIOQ values, these vary around 0.0012. Furthermore, TRIQ values are stable for all solutions except TRI₉−TRI₁₀ due to the genetic algorithms nature. In conclusion, TRI₉ is the best solution since it has the best value of TRIQ, closely followed by TRI₁₀.

Table 2 MSR_3D Elutriation solution table

Elutriation LSL experiment

In Table 3 you can see how TRI₃ has the best value of BIOQ, TRI₂ has the best value of GRQ, TRI₆ has the best value of PEQ and, TRI₁ has the best value of SPQ. In general, the GRQ, PEQ and SPQ values vary around an average value from TRI₁ until TRI₈. Then, these values decrease in TRI₉−TRI₁₀ solutions due to the fact that the algorithm reached a local minimum in this two solutions; the BIOQ values fluctuate around 0.0012 value reaching a maximum in TRI₃ and a minimum in TRI₄. The values of TRIQ reach the maximum values at the first two solutions, then remain stable and finally fall in local minimum in the last two solutions. In conclusion, TRI₁ is the best solution since it has the best value of TRIQ.

Table 3 LSL Elutriation solution table

Elutriation MSL experiment

We can observe in Table 4 how TRI₂ has the best value of BIOQ, PEQ and SPQ whereas TRI₁ has the best value of GRQ. The GRQ, PEQ and SPQ have a stable fluctuation throughout the solutions whilst BIOQ varying around the central value 0.0011. The TRIQ values reach their maximum value at TRI₂, the minimum at TRI₃ and the rest are stabilized. In conclusion, TRI₂ is the best solution since it has the best value of TRIQ.

Table 4 MSL Elutriation solution table

Elutriation OPT experiment

We can verify in Table 5 how all triclusters have the same value of BIOQ since all triclusters grouped the same collection of genes. Regarding GRQ index, the triclusters have values between 0.70 and 0.86 with the exception of TRI₁, TRI₉ and, TRI₈ being TRI₄ the solution with better GRQ. The PEQ and SPQ indexes have fluctuating values being TRI₇ the tricluster with the better PEQ and SPQ. In conclusion, TRI₇ is the best solution since it has the best value of TRIQ.

Table 5 OPT Elutriation solution table

Elutriation summary

We can see in Fig. 3 how the solutions are distributed regarding BIOQ and GRQ for each experiment. We observe that all points are concentrated in a BIOQ interval of [0.000728,0.0013] for each experiment meanwhile the MSL experiment stands out because all its solutions have a GRQ near to 1. Regarding the PEQ and SPQ solutions distribution, we can see in Fig. 4 how the majority of the solutions are concentrated around the point PEQ=0.325,SPQ=0.325 in the MSR_3D experiment, all solutions are concentrated in [0.50,0.75] interval for PEQ and SPQ in the LSL experiment, all solutions are concentrated in [0.75,1.00] interval for PEQ and SPQ in the MSL experiment and, all solutions are concentrated in [0.30,0.70] interval for PEQ and SPQ in the OPT experiment.

Fig. 3

BIOQ vs GRQ dispersion graph for each Elutriation solution of each experiment

Fig. 4

PEQ vs SPQ dispersion graph for each Elutriation solution of each experiment

The global TRIQ-based ranking of solutions is showed in Table 6; we can see how the solutions of the MSL experiment are placed on the first positions followed by two outstanding solutions of the MSR_3D experiment, all solutions of the LSL experiment, all solutions of the OPT experiment and, the remaining solutions of the MSR_3D experiment.

Table 6 Elutriation ranking table

The MSL experiment has the best average values of TRIQ and the lowest standard deviation of TRIQ as seen in Table 7. This fact is reflected in Fig. 5 wherein the MSL point is located on the bottom-right side of the graph which implies that the MSL experiment has the highest values of TRIQ and a sparsely dispersed distribution, thus this is a high-quality experiment.

Fig. 5

MEAN vs STDEV dispersion graph for each Elutriation experiment

Table 7 Elutriation summary table

The most valuable solution of all experiments is the tricluster TRI₂ of the MSL experiment. We can see in Fig. 6 its three graphic views showing that its high value of GRQ is reflected in the patterns depicted. Furthermore, in Table 8 we observe terms with moderately low p-value as fermentation, vesicle fusion to plasma membrane and exocytosis. Fermentation is a biological process that is part of the process called energy derivation by oxidation of organic compounds and, in turn, belongs to the generation of precursor metabolites and energy process and the oxidation-reduction process; Vesicle fusion to plasma membrane is a biological process that is part of the exocytosis proccess; the first term is a process of cellular component organization whereas the second is an establishment of localization process.

Fig. 6

TRI₂ graphic views of the Elutriation MSL experiment

Table 8 TRI₂ GO table of the MSL Elutriation experiment

Mouse GDS4510 dataset

This batch corresponds to the mouse GDS4510 dataset. This dataset was obtained from GEO [35] with accession code GDS4510 and title rd1 model of retinal degeneration: time course [33]. In this experiment, the degeneration of retinal cells in different individuals of home mouse (Mus musculus) is analyzed over 4 days just after birth, specifically on days 2, 4, 6 and 8.

For our purpose, we have created a dataset $D_{GDS4510_{3D}}$ composed of 22690 genes, 8 experimental conditions (one for each individual involved in the biological experiment) and 4 time points.

$D_{GDS4510_{3D}}$ has been used as the input of the TriGen and the OPTtricluster algorithm in four experiments: MSR_3D, LSL, MSL and, OPT.

GDS4510 M S R _3D experiment

We can verify in Table 9 how TRI₇ has the best value of BIOQ, GRQ, PEQ, SPQ. The GRQ, PEQ and SPQ indexes vary uniformly among all the solutions. BIOQ has a peak of TRI₇ which has the maximum value. The TRIQ values oscillate between 0.385 and 0.4 with the exception of TRI₇, therefore this is the best solution since it has the best value of TRIQ.

Table 9 MSR_3D GDS4510 solution table

GDS4510 LSL experiment

In Table 10 we can see how TRI₁ has the best value of BIOQ and GRQ meanwhile TRI₂ has the best values of PEQ and SPQ. The GRQ, PEQ and SPQ values vary uniformly around a central value among the triclusters whereas BIOQ has peak values in TRI₁ and TRI₄. The TRIQ values oscillates between 0.40 and 0.43 being TRI₁, TRI₄ and TRI₉ the most outstanding solutions. We can conclude that TRI₁ is the best solution since it has the best value of TRIQ.

Table 10 LSL GDS4510 solution table

GDS4510 MSL experiment

For this experiment, we can observe in Table 11 how TRI₁ has the best value of BIOQ and GRQ meanwhile TRI₂ has the best value of PEQ and TRI₈ has the best value of SPQ. The PEQ and SPQ indexes of all solutions vary uniformly around 0.5 whereas all the GRQ values are close to 0.9. The BIOQ values oscillate between 0.0012 and 0.0019 reaching its higher value in the TRI₁ solution. The TRIQ values are in the [0.42,0.44] interval, therefore we can conclude that they are good results for this experiment. The highest value of TRIQ is reached by TRI₁, hence it is the best solution for this experiment.

Table 11 MSL GDS4510 solution table

GDS4510 OPT experiment

In Table 12 we can see how TRI₂ has the best value of BIOQ, TRI₄ has the best value of GRQ and, TRI₉ and TRI₁ have the best value of PEQ and SPQ respectively. The BIOQ values vary among [0.0012,0.0016] interval with the exception of TRI₂ and TRI₃ whilst the GRQ values vary uniformly around the 0.80 value excepting TRI₄. The PEQ and SPQ values oscillate among the [0.5,0.8] interval.The highest value of TRIQ is reached by TRI₄, thus it is the best solution for this experiment.

Table 12 OPT GDS4510 solution table

GDS4510 summary

We can see how the solutions are distributed regarding BIOQ and GRQ in Fig. 7; we observe that all points of all experiments are concentrated in a BIOQ interval of [0.0011,0.0059]. Regarding the GRQ values, the MSR_3D and LSL experiments have all the solutions in the [0.83,0.90] interval, the MSL experiment has all the solutions in the [0.92,0.99] interval and, the OPT experiment has all the solutions in the [0.80,0.95] interval. Regarding the PEQ and SPQ distribution we can see in Fig. 8 how the majority of solutions are concentrated around the point PEQ=0.5,SPQ=0.5 in the MSR_3D and MSL experiments, meanwhile the solutions of LSL experiment are concentrated in the interval [0.625,0.75] for PEQ and SPQ values and, the OPT experiment has his solutions dispersed in two groups: one group around the PEQ=0.5,SPQ=0.5 point and the other in an interval of [0.60,0.83] for both PEQ and SPQ values.

Fig. 7

BIOQ vs GRQ dispersion graph for each GDS4510 solution of each experiment

Fig. 8

PEQ vs SPQ dispersion graph for each GDS4510 solution of each experiment

A global TRIQ-based ranking of solutions is shown in Table 13. The MSL, LSL and a part of OPT solutions are placed alternatively on the first positions and the MSR_3D and the remaining of OPT solutions are in the last positions.

Table 13 GDS4510 ranking table

We can see in Table 14 how the GDS4510 MSL experiment has the best value of the mean of TRIQ and the four experiments have low values of standard deviation having the MSR_3D experiment the lowest value but very close to the MSL one. This fact implies that the four experiments have a low sparse distribution and solutions with high quality. We can see in Fig. 9 how the MSR_3D, LSL and, MSL points are located on the bottom side of the graph meanwhile the OPT point is located in a high level of the standard deviation axis; on the other hand, LSL and, MSL points are located on the right side of the graph meanwhile the MSR_3D and OPT points are located in a left level of the average axis. Hence, in terms of standard deviation and average, we can conclude that MSL is the best experiment.

Fig. 9

MEAN vs STDEV dispersion graph for each GDS4510 experiment

Table 14 GDS4510 summary table

The most valuable solution of all experiments is the tricluster TRI₁ of the MSL experiment. We can see in Fig. 10 how this solution depicts very uniform patterns consistent with the GRQ value. Also, we can see in Table 15 that this solution has Gene Ontology terms with low p-value such as sensory perception of chemical stimulus, olfactory receptor activity or detection of chemical stimulus involved in sensory perception of smell. The term olfactory receptor activity is a molecular function that combining with an odorant and transmitting the signal from one side of the membrane to the other to initiate a change in cell activity in response to detection of smell; this function is part of the biological process detection of chemical stimulus involved in sensory perception of smell that is the series of events involved in the perception of smell in which an olfactory chemical stimulus is received and converted into a molecular signal. Finally, that process is framed in a more general biological process called sensory perception of chemical stimulus that is the series of events required for an organism to receive a sensory chemical stimulus, convert it to a molecular signal, and recognize and characterize the signal.

Fig. 10

TRI₁ graphic views of the GDS4510 MSL experiment

Table 15 TRI₁ GO table of the MSL GDS4510 experiment

Human GDS4472 dataset

The dataset, corresponding to this batch, has been obtained from GEO [35] under code GDS4472 titled Transcription factor oncogene OTX2 silencing effect on D425 medulloblastoma cell line: time course [34]. In this experiment, the effect of doxycycline on medulloblastoma cancerous cells at six times after induction (0, 8, 16, 24, 48 and 96 h) had been studied.

Our input dataset $D_{GSD4472_{3D}}$ is composed of 54675 genes, 4 conditions (one for each individual involved) and 6 time points (one per hour) and has been used as the input of the TriGen and the OPTtricluster algorithm in four experiments: MSR_3D, LSL, MSL and, OPT.

GDS4472 M S R _3D experiment

For this experiment, TRI₄ has the best value of BIOQ, TRI₆ has the best value of PEQ, TRI₃ has the best value of SPQ and TRI₅ has the best value of GRQ as you can see Table 16. The PEQ and SPQ values of the solutions oscillate around 0.64 and the GRQ values vary between 0.76 and 0.64; the BIOQ index oscillates around 0.0014 reaching two peaks at TRI₄ and TRI₈. In general, the TRIQ value of solutions are in [0.32,0.37] having TRI₃ and TRI₇ as outstanding ones and TRI₅ as the best solution in this experiment.

Table 16 MSR_3D GDS4472 solution table

GDS4472 LSL experiment

We can verify in Table 17 how TRI₁ has the best values of BIOQ, GRQ, PEQ and SPQ. In general, the GRQ, PEQ and SPQ indexes of the solutions depicts homogeneous values with the exception of TRI₁ where they reach their maximum; regarding BIOQ values, those reach three peaks at TRI₁, TRI₄ and TRI₁₀. The TRIQ values vary between 0.39 and 0.44 being TRI₁ the best solution of this experiment.

Table 17 LSL GDS4472 solution table

GDS4472 MSL experiment

In Table 18 we can see how TRI₉ has the best values of BIOQ and GRQ while TRI₇ has the best value of PEQ and TRI₁₀ has the best value of SPQ. The PEQ values of the solutions vary in the [0.43,0.46] interval and the SEQ values are in the [0.40,0.44] interval while all solutions have high GRQ values close to 0.90; the BIOQ values have three peaks at TRI₅, TRI₇ and TRI₉. Regarding TRIQ values, they vary in [0.40,0.42] interval being TRI₁, TRI₅ and TRI₇ the outstanding solutions and being TRI₉ the best solution.

Table 18 MSL GDS4472 solution table

GDS4472 OPT experiment

For this experiment, TRI₅ has the best value of BIOQ, TRI₁₀ has the best value of GRQ and SPQ and, TRI₈ has the best value of PEQ as you can see Table 19. The BIOQ index oscillates around 0.0015 reaching three peaks at TRI₅, TRI₉ and, TRI₁₀. The GRQ index vary in the [0.6,07] interval reaching an outstanding value in the TRI₁₀ solution. Regarding the PEQ values they vary in a interval of [0.42,0.86] and the SPQ values in the [0.34,0.76] interval. The TRIQ values vary between 0.28 and 0.44 being TRI₁₀ the best solution of this experiment.

Table 19 OPT GDS4472 solution table

GDS4472 summary

We can observe in Fig. 11 how the solutions of the four experiments are in a BIOQ interval of [0.0012,0.0272] meanwhile the GRQ values of the solutions of MSR_3D are in the [0.6451,0.7615] interval, the solutions of LSL are in the [0.8623,0.8953] interval, the solutions of MSL are in the [0.8964,0.9238] interval and, the solutions of OPT are in the [0.6,0.7] interval with an outstanding point near to GRQ=0.92. Regarding the PEQ and SPQ solutions distribution we can see in Fig. 12 how the PEQ and SPQ of MSR_3D are concentrated in the [0.50,0.75] interval, the values PEQ and SPQ of LSL are in the [0.325,0.75] interval, the values PEQ and SPQ of MSL are in the [0.325,0.50] interval and, the values PEQ and SPQ of OPT are dispersed in three groups: the first in the [0.42,0.45] interval for PEQ and SPQ, the second in the [0.70,0.85] interval for PEQ and the [0.46,0.54] interval for SPQ and the third, that is a single point, in PEQ=0.74,SPQ=0.76.

Fig. 11

BIOQ vs GRQ dispersion graph for each GDS4472 solution of each experiment

Fig. 12

PEQ vs SPQ dispersion graph for each GDS4472 solution of each experiment

We can see the global TRIQ-based ranking of solutions in Table 20; the MSL solutions, one OPT solution and, the LSL solutions are placed alternatively on the first positions and the MSR_3D and the remaining of OPT solutions are on the last positions.

Table 20 GDS4472 ranking table

We can see in Table 21 how the MSL experiment has the best value of the average and standard deviation of TRIQ, however, the LSL experiment has the best tricluster closely followed by the OPT experiment. In Fig. 13 we can see how the MSL is placed in the bottom-right position being the best experiment in terms of standard deviation and average.

Fig. 13

MEAN vs STDEV dispersion graph for each GDS4472 experiment

Table 21 GDS4472 summary table

The most valuable solution of all experiments is the tricluster TRI₁ of the LSL experiment. This solution depicts very uniform patterns since has a very high GRQ value, we can check this fact in Fig. 14. Also, we can see in Table 22 that this solution has Gene Ontology terms with very low p-value such as SRP-dependent cotranslational protein targeting to membrane, nuclear-transcribed mRNA catabolic process, nonsense-mediated decay or ribonucleoprotein complex.

Fig. 14

TRI₁ graphic views of the GDS4472 LSL experiment

Table 22 TRI₁ GO table of the LSL GDS4472 experiment

The SRP-dependent cotranslational protein targeting to membrane process is described as the targeting of proteins to a membrane that occurs during translation and is dependent upon two key components, the signal-recognition particle (SRP) and the SRP receptor. SRP is a cytosolic particle that transiently binds to the endoplasmic reticulum (ER) signal sequence in a nascent protein, to the large ribosomal unit, and to the SRP receptor in the ER membrane; it is a protein targeting process that occurs in the intracellular component and is part of the cellular protein localization process. The nuclear-transcribed mRNA catabolic process, nonsense-mediated decay is a biological process that describes the nonsense-mediated decay pathway for nuclear-transcribed mRNAs degrades mRNAs in which an amino-acid codon has changed to a nonsense codon; this prevents the translation of such mRNAs into truncated, and potentially harmful, proteins; it is a negative regulation of gene expression process that negatively regulates the macromolecule metabolic process. Finally the ribonucleoprotein complex is a cellular component that is defined as a macromolecular complex containing both protein and RNA molecules.

Conclusions and discussion

Although triclustering has emerged as an essential task to study 3D datasets, there is no consensus on how to evaluate tricluster solutions obtained from each data set. Different authors validate their triclusters on different measures, with correlation, graphic validation and Gene Ontology terms being the most common ones. In this work we have presented a tricluster validation measure, TRIQ, a single evaluation measure that combines the information from the three aforementioned sources of validation.

We have applied TRIQ to three different datasets: the yeast cell cycle (Saccharomyces Cerevisiae), in particular the elutriation experiment, an experiment with mice (Mus Musculus) called GDS4510 and data from an experiments with humans (Homo Sapiens) called GDS4472.

We have shown that TRIQ has successfully resumed the three validation measures (correlation, graphic validation and Gene Ontology terms) yielding the same validation results as in [27] where each of the components of TRIQ (BIOQ, GRQ, PEQ, and SPQ) where applied separately. In that publication we presented the MSL measure, comparing it to MSR_3D and LSL, with the same datasets used in this article. We concluded that MSL was the best fitness function. In this publication, we have seen how MSL has obtained the best general results, with high values of TRIQ and low standard deviation for all solutions presented. Therefore, we can conclude that TRIQ has been successful in representing and summarizing the individual values provided by BIOQ, GRQ, PEQ, and SPQ. Furthermore, we have applied TRIQ to results from another algorithm, OPTRicluster, and we have shown how TRIQ has been a valid tool to compare results from different algorithms in a quantitative straightforward manner.

For the case of triclustering being applied to not biologically related fields as in [36], TRIQ can also cope with the analysis of the tricluster solutions thanks to the weighting system (see “Methods” section), which allows for each term to be included or removed in the final measure.